# Point Groups

Point groups are a method of classifying the shapes of molecules according to their symmetry elements.

Before we can talk about point groups, we need to describe the basic elements of symmetry. These are the proper axis of symmetry (or just axis of symmetry), improper axis of symmetry, plane of symmetry, and inversion center (or point of symmetry).

• The proper axis of symmetry is an imaginary line through a compound. Rotation of the compound by an integral fraction of a circle around this axis (1/2, 1/3, etc.) brings the compound to superposition on itself. An axis of symmetry is represented by Cn, where n is the integral fraction. All shapes have at least a C1 axis (the identity axis).

• The improper axis of symmetry is also an imaginary line through a compound. Rotation of the compound by an integral fraction of a circle around this axis (1/2, 1/4, etc.) followed by reflection through a plane perpendicular to this axis brings the compound to superposition on itself. An improper axis of symmetry is represented by Sn, where n is the integral fraction (n must be even or 1).

• The plane of symmetry is a plane through a compound that relates its identical halves. A plane of symmetry is represented by σ and is the same as an S1 axis.

• The inversion center (or point of symmetry) is an imaginary point in a compound. Reflection of the compound through this point brings the compound to superposition on itself. An inversion center is represented by i and is the same as an S2 axis.

The main classes of point groups are C, D, S, T, O, and I. The first two classes are most common. Each of these classes is subdivided into different point groups.

• Compounds in the C class can be Cs, Ci, Cn, Cnv, or Cnh, where n is an integer.

• Compounds in the Cs point group have a plane of symmetry and nothing more.

• Compounds in the Ci point group have a center of inversion and nothing more.

• Compounds in the Cn point group have exactly one Cn axis and nothing more. Compounds in this point group are always chiral. Examples.

Compounds in the C1 point group have only the identity axis; in other words, they have no symmetry. Most compounds have C1 symmetry.
• Compounds in the Cnv point group have a Cn axis and n planes of symmetry containing the Cn axis. They do not have a plane of symmetry perpendicular to the Cn axis. Examples.

Linear molecules with conical symmetry (CO, ClC≡CH) are classified in the C∞v point group.
• Compounds in the Cnh point group have a Cn axis and a plane of symmetry perpendicular to the Cn axis. Examples.

• Compounds in the D class contain a Cn axis and n C2 axes perpendicular to it. Compounds in the D class can be Dn, Dnd, or Dnh, where n is an integer.

• Compounds in the Dn point group have no additional elements of symmetry. Compounds in this point group are always chiral. This point group is not common. Examples.

• Compounds in the Dnd point group have n planes of symmetry containing the Cn axis. They do not have a plane of symmetry perpendicular to the Cn axis. Examples.

• Compounds in the Dnh point group have a plane of symmetry perpendicular to the Cn axis. Examples.

Linear molecules with cylindrical symmetry (N2, HC≡CH) are classified in the D∞h point group.
• Compounds in the S class are Sn, where n is an even integer. Compounds in the S class contain a Sn axis plus a Cn/2 axis coinciding with it.

• Compounds in the T (tetrahedral) class can be T, Td, or Th. Compounds in the T class contain four C3 axes and three C2 axes.

• Compounds in the T point group have no additional elements of symmetry. Compounds in this point group are always chiral. This point group is not common. The compound C(XS)4, where XS is a group with a stereocenter with the (S) configuration, is in this point group.

• Compounds in the Td point group have three planes of symmetry that contain the C2 axes. This point group is not common. An octasubstituted cubane in which each vertex is alternately substituted with XR and XS is in this point group.

• Compounds in the Th point group have six planes of symmetry that contain the C3 axes. Compounds such as CH4 are in this point group.

• Compounds in the O (octahedral) and I (icosahedral) classes have even higher symmetry. These point groups are not common. PF6 is in the Oh point group, and C60 is in the Ih point group.

You should also remember that most compounds are conformationally mobile (i.e., constantly changing their shape), and, as a result, the point group of a compound can depend on the time scale. For example, at a very short time scale, 1-propyne (HC≡CCH3) is in the C3v point group, but at longer time scales, rapid rotation about the C–CH3 bond puts it in the C∞v point group. Cyclohexane is in the D3d point group at short time scales, when it is in a single chair form, but at longer time scales, at which it is in rapid equilibrium between its two chair forms, it is in the D6h point group.